Grain-Size Distribution in Metals and Its Influence on Ultrasonic Attenuation Measurements

A transformation has been derived relating the number of spheres of a certain radius R per unit volume (the “volume distribution of spheres”) to the number of circles smaller than a certain radius r per unit area (the “area distribution of circles”) appearing on a plane cutting through the volume. The transformation was applied to several hypothetical grain-size distributions for polycrystalline metals to find the resulting hypothetical area distribution of grain images on photomicrographs. Comparison of the hypothetical area distributions to experimentally found area distributions gave the following conditions that the true volume distribution of grains must meet: (1) It must be finite at R=0, and (2) it must have a nonzero decreasing tail for large values of R. The common assumption of a single grain diameter is insufficient to explain the experimental area distribution of grain images. The functions NV(R) = Rn exp(−kR) and NV(R) = exp[−(lnR/R0)2/2σV2] were judged plausible for the volume distribution function of grains, and a correction was computed for the attenuation formulas for Rayleigh scattering of ultrasonic waves in polycrystalline metals by taking averages of R6 and R3 over these functions.